Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal. Let $R(\mathfrak{a}) = \oplus_{n\geq 0} \mathfrak{a}^n$ be the Rees algebra of $\mathfrak{a}$. Let $I$ and $J$ be two graded ideals of $R(\mathfrak{a})$. Let $m, n$ and $p$ be the maximal degrees of generators of $I, J$ and $I \cap J$, respective.

Question: Can we bound $p$ in term $m$ and $n$? i.e. Does there exist a function $f(x, y)$ such that $p \leq f(m,n)$ for all ideals $I$ and $J$?